The question in the image involves the functions $f(x)$ and $g(x)$, where the following details are provided:

1. $f(x) - 3g(x) = 2x^4 + 3x^3 + 6x + 4$
2. $f(x)$ is an even function, and $g(x)$ is an odd function.

We are tasked to find the value of $f(1) + g(1)$.

Step 1: Using the properties of even and odd functions

• For $f(x)$ (even function): $f(-x) = f(x)$.
• For $g(x)$ (odd function): $g(-x) = -g(x)$.

This means that the behavior of $f(x)$ and $g(x)$ will help simplify calculations.

Step 2: Substitute $x = 1$ and $x = -1$ into the given equation

The equation is: $f(x) - 3g(x) = 2x^4 + 3x^3 + 6x + 4$

For $x = 1$:

$f(1) - 3g(1) = 2(1)^4 + 3(1)^3 + 6(1) + 4$ $f(1) - 3g(1) = 2 + 3 + 6 + 4 = 15$

For $x = -1$:

$f(-1) - 3g(-1) = 2(-1)^4 + 3(-1)^3 + 6(-1) + 4$ $f(-1) - 3(-g(1)) = 2 - 3 - 6 + 4 = -3$

Simplify: $f(-1) + 3g(1) = -3$

Step 3: Use the even and odd properties

From the properties of $f(x)$ and $g(x)$:

• $f(-1) = f(1)$ (since $f(x)$ is even),
• $g(-1) = -g(1)$ (since $g(x)$ is odd).

Substitute $f(-1) = f(1)$ into the second equation: $f(1) + 3g(1) = -3$

Step 4: Solve the system of equations

We now have two equations:

1. $f(1) - 3g(1) = 15$,
2. $f(1) + 3g(1) = -3$.

Add the two equations: [ [f(1) - 3g(1)] + [f(1) + 3g(1)] = 15 + (-3) ] $2f(1) = 12 \implies f(1) = 6$

Substitute $f(1) = 6$ into the first equation: $6 - 3g(1) = 15$ $-3g(1) = 9 \implies g(1) = -3$

Step 5: Calculate $f(1) + g(1)$

$f(1) + g(1) = 6 + (-3) = 3$

Final Answer:

The correct choice is B) 3.

Do you want further clarification or detailed explanations?
Here are 5 related questions to deepen understanding:

1. What is the difference between even and odd functions?
2. How do you identify even and odd functions from their equations?
3. Can a function be both even and odd? Why or why not?
4. How does symmetry affect the evaluation of functions at specific points?
5. What would happen if both functions in this problem were even or odd?

Tip: When solving equations involving symmetry, always exploit the properties of even and odd functions to simplify calculations.